A class of strongly nonlinear functional differential equations
type: Item
instance of: scholarly article
title: A class of strongly nonlinear functional differential equations
zbMATH Open document ID: 0667.34080
DOI: 10.1007/BF01762791
author: Enzo Mitidieri
author: Ioan I. Vrabie
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1988-01-01T00:00:00Z
review text: The existence of solutions of the functional differential equation \[ u'(t)+\partial \phi (u(t))\in F(t,u(t),u_ t),\quad 0<t<T,\quad u(s)=v(s),\quad -\tau \leq s\leq 0 \] is considered, where \(\partial \phi\) is the subdifferential of \(\phi\). Existence is proved by a fixed point argument based on the result that a function P: \(C\to C\), where C is a nonempty, convex and weakly compact subset of a separable locally convex space, which is weakly sequentially continuous has at least one fixed point. A theorem of Brézis shows that the equation \(u_ h'(t)=\partial \phi (u_ h(t))\ni h(t),\) \(u_ h(0)=v(0)\) has a solution for h in a certain ball in \(L^ 2\). Then the existence theorem follows by defining P by \((Ph)(t)=F(t,u_ h(t),\hat u_{h_ t}),\) and placing a certain demiclosedness condition on F. Here, \(\hat u(t)=v(t)\) for \(t\in [- \tau,0]\), \(\hat u(t)=u(t)\) for \(t\in (0,T_ 0]\), continuation of solutions is proved in the usual way and a number of examples are presented, including the Navier-Stokes equations with a delay in the viscosity.
Mathematics Subject Classification ID: 34K05
zbMATH DE Number: 4091956
zbMATH Keywords: functional differential equation
zbMATH Keywords: continuation of solutions
zbMATH Keywords: examples
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: viscosity
reviewed by: S. P. Banks
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01762791
OpenAlex ID: W1965365481
description: scientific article; zbMATH DE number 4091956